We now can switch the order of max and min via Sion's minimax theorem and maximize over \(\mathbf {V}_i\) . Defining \(\mathbf {K}_{c, s}\)  as the \((c, s)\)  commutation matrix [1]}:
\(\mathbf {vec}({\mathbf {X}}_i \otimes \mathbf {V}_i)= \left((\mathbf {I}_{d} \otimes \mathbf {K}_{c, s})(\mathbf {vec}({\mathbf {X}}_i) \otimes \mathbf {I}_c) \otimes \mathbf {I}_{s}\right)\mathbf {vec}(\mathbf {V}_i)\) 
